Materials for Biomedical Engineering. Mohamed N. Rahaman
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We take the reciprocals of the ratios of the intercepts to the corresponding unit cell dimensions
Then the numbers are converted to the smallest integers
As an example, for a cubic crystal structure illustrated in Figure 3.25, the plane A′B′C′ has intercepts OA′ = 2a, OB′ = 3a, and OC′ = 3a. The reciprocals of the ratios of the intercepts to the corresponding cell dimensions are respectively
Converting these numbers to the smallest integers, we get 3, 2, and 2, respectively. These numbers are written in parentheses (322) and they are called the Miller indices of the plane. Thus, the plane A′B′C′ is referred to as the (322) plane. In general, a plane with Miller indices h, k, and l is written as (hkl).
A negative intercept of a plane on a crystal axis is written with a bar over the respective index. A plane that has intercepts of 2a, 3a, and −3a, respectively, on the x, y, and z axes, for example, has Miller indices of (32
Depending on the crystal system, two or more planes may belong to the same family of planes. In the cubic system, for example, the six sides of the unit cell has Miller indices of
These planes are equivalent except for our arbitrary choice of axis labels and directions, or, alternatively, on how we view the orientation of the unit cell in space. They have the same arrangement of atoms and, thus, the properties in these planes are identical. These planes are described collectively as {100} planes where the Miller indices are enclosed within curly brackets. In general, a family of equivalent planes with Miller indices h, k, and l is written {hkl}.
Lattice Directions
Any direction in a crystal lattice (Figure 3.26) is described by:
A line through the origin O of the unit cell that is parallel to the given direction;
Resolving the length of this line along the three principal axes x, y, and z and expressing these lengths as a fraction of the unit cell dimensions a, b, and c;
Converting the ratios to the three smallest integers, u, v, and w, for example
Figure 3.26 Diagram illustrating the specification of lattice directions in a crystal.
The convention used to describe the direction is to write the three integers u, v, and w in square brackets, that is, [uvw]. The direction LM in Figure 3.26, for example, is found by drawing the line OE through the origin that is parallel to it. Resolving the length of OE along the x, y, and z axes and expressing them as a fraction of the unit cell dimensions, we get
Thus, the direction LM is described as [111], as are directions parallel to LM or OE.
Similar to planes, we can have families of equivalent directions. In the cubic system, for example, the directions
are equivalent except for our arbitrary choice of axis labels and directions. They are denoted collectively as 〈111〉 directions where the integers are enclosed within angle brackets. In general, a family of equivalent directions described by the integers u, v, and w is written 〈uvw〉. Properties of a cubic crystal that depend on direction, such as the elastic modulus, for example, will be identical in these eight directions.
3.7 Concluding Remarks
In this chapter, we discussed how atoms pack in three dimensions to form solids, defects in crystalline solids and microstructure of materials, which, along with atomic bonding (Chapter 2), are the structural features that control the properties of a material.
Packing of atoms in solids is divided into two broad classes, crystalline, and amorphous.
The vast majority of metals and ceramics are not composed of a single crystal but, instead, are polycrystalline, composed of a large number of small crystals called grains. Atomic packing defects composed of point defects, line defects (dislocations) and grain boundaries exert a strong effect on the physical and mechanical properties of crystalline solids.
Inorganic glasses are amorphous but when subjected to a controlled heat treatment, some glasses can be converted to glass‐ceramics, which often have better properties than their parent glasses.
The majority of polymers are amorphous but under appropriate conditions, some polymers develop a semicrystalline structure composed of crystalline and amorphous regions. Whether a polymer is amorphous or semicrystalline has a strong influence on their properties.
We discussed more recently discovered allotropes of carbon, such as graphene, carbon nanotubes, and fullerenes, which are receiving considerable interest for use in biomedical applications due to their unique properties.
As the engineering properties of materials are dependent on their microstructure, the ability to control the microstructure of a biomaterial is important for achieving a desired combination of properties for a particular application.
Problems
1 3.1 Aluminum has an atomic radius of 0.1431 nm and an FCC structure. Determine (i) the number of atoms in a unit cell, (ii) the unit cell length, and (iii) the density of aluminum. (Atomic weight of aluminum = 26.97; Avogadro number = 6.023 × 1023.)
2 3.2 Determine the atomic packing fraction for the simple cubic, BCC and FCC structures composed of a single type of atom. Explain the steps in your calculation.
3 3.3 A metal has an atomic radius r and a CPH structure. Determine the unit cell lengths in terms of r and the atomic packing fraction. Explain the steps in your calculation.
4 3.4 Using the unit cell illustrated in Figure 3.10 and the unit cell lengths a = b = 0.9430 nm and c = 0.6891 nm, determine the density of hydroxyapatite.
5 3.5 The use of fluorinated water or toothpaste is known to improve the resistance of teeth to decay. Can this be accounted for in terms of the structure of hydroxyapatite. Explain.
6 3.6 Some drug delivery devices are composed of drug molecules incorporated within a degradable polymer. As the polymer degrades, these molecules are released at a controllable